III.I.

There are times when, for any number of personal or professional
reasons, two mathematicians
with a longstanding collaboration can no longer work together. They
must go through the difficult
process of determining how to complete and publish their joint
results. Here are a few stories
regarding mathematical divorces that I have gathered for the purposes
of the panel. None of
these stories concern mathematicians in my own field, but they are
stories of mathematicians in
fields in which the publication of 1-2 major papers per year is
considered to be a successful rate of output.
The
loss of a single such paper can therefore drastically effect tenure, promotion
and NSF funding.

The following stories were told to me from the perspective of
Mathematician A.

Story 1:

Mathematician A and B are having a falling out because Mathematician B
believes he or she has all the
good ideas. Mathematician A feels that he/she is being treated
as the one who merely verifies and types up the
results. Mathematician A
is particularly upset because his or her ideas would complete the project
sooner and there are other projects demanding attention.
Both A and B are young and are concerned about promotion and
jobs. There is a senior mathematician
that both respect. Mathematician A explains his/her concerns to this senior
mathematician, and can do so without risking any accusations which would
have a negative impact on B.
The mediator has sufficient expertise in the field to evaluate the merit of A's suggestions and
explicitly advises B to
apply A's suggestion. Mathematician B does this,
the paper is completed, and they
never coauthor again.

Note that in this story, the fact that Mathematician B doesn't trust
other people's ideas has not been
addressed. On the one hand, it was not necessary to deal with
this larger issue in order to mediate the problem and to allow both A and B to complete
their paper. On the other hand, while this served A very well, mediation may have
failed to address a problem which will affect B's future mathematical relationships.

Story 2:

Mathematician A and B have a multiyear project with no completed
papers but many theorems.
Both mathematicians are equally stressed about promotion and have been
presenting their joint work
without producing preprints. The delay in the production of
preprints is due to the fact that neither
mathematician really trusts the other mathematician's proofs. They are in
different fields and cannot verify
each other's work. Each has begun accusing the other of not being
rigorous. As an outsider judging
these subfields, I would say that they are characterized by an abundance of tersely written papers
for an audience of experts. Rather than mutually agreeing that
they each need to put in the extra
effort to make their work intelligible to both fields: these two
mathematicians have reached the point of directly attacking one another personally. Finally
Mathematician B declares
that the research need never appear in print.

Both mathematicians complain to their doctoral advisors but they do
not find a common mediator. The
advisors tell each author that this work is important, must appear in
print, and that both names should appear
on all work produced from this project. Mathematician A's advisor
suggests trusting Mathematician B's ability
to write a proof. Mathematician B's advisor suggests Mathematician B
write up everything he or she can prove independently
and make that a joint paper but not to trust Mathematician A. Both
mathematicians follow their
respective advisor's advise, but the
conflicting advise does not, of course, help to resolve the impasse.

Mathematician B writes a paper containing some of their joint work,
which A agrees to co-author. The
paper is submitted to, and quickly accepted by, a journal in Mathematician B's field.
After A writes up the rest of the results, there follows a long period of heated arguments while
B claims to be checking the paper, and A's write up is finally submitted and accepted for publication.
In the end, a sufficient number of the results are published, and they papers have
a tremendous positive impact on each of A's and B's careers. But the two are no longer
on speaking terms and some of
their results may not ever be published. It is evident from the publication record that
both A and B should now trust one another to write up a coherent account of the mathematics, but
they are, unfortunately, unable to reach an understanding.

To their credit, these mathematicians have not publicly complained
about their coauthoring troubles. It is
immediately obvious to the mathematical community who wrote up which
paper, but this is not seen as a sign
of a problem. Coauthors will often divide results according to
subfields and publish in specialist journals. In
fact, these two might have avoided the divorce in the first place, if
they had agreed on this approach right from the start.

Story 3:

Mathematician A and B have a multi-year project which has not yet
produced any papers. Mathematician A is tenured, with no
looming deadlines for grants
or promotion and finds him(her)self unable to continue the collaboration in
a timely fashion. He(she) decides to bring the projects to a close.
Mathematician B, however, is preparing for promotion to tenure and wants to continue
the joint work. Mathematician A proposes the following solution: A will essentially abandon authorship claim to
the part of the project which is nearest to completion and pursue the longer term aspects of the project later on his(her) own. Mathematician B agrees to the plan and completes the first paper with a new coauthor. Then B even offers A co-authorship
as a gesture of good will, which Mathematician A declines. So they just cite each
others' work and acknowledge each other for important ideas leading to
the results.

This story ends with the two mathematicians on good terms and able to
communicate with one another
and support each other's work. Any divorce of a senior
mathematician from a junior mathematician
can be stressful to the junior mathematician. Here, the senior mathematician
has chosen to be generous and sensitive to the timing issues of
the junior mathematician, even while being clear that the joint work had come to an end.
The choice to abandon the project could have happened for any number of non-mathematical reasons, including health issues, relocation, and choices about prioritizing work.

Closing Remarks:

Finally, I would like to mention a question that was asked on the live
panel at the Joint AMS meetings from a
young mathematician had been through three mathematical divorces in a
row and wondered why. Most likely this
person was exceptionally unlucky. However, it is worth discussing the
scenarios in more detail with someone to
look for a common theme. It is always helpful to step back,
after a mathematical divorce, and ask if there was anything that could
have been done differently.